Dynamic Longevity Hedging in the Presence of Population Basis Risk:

A Feasibility Analysis from Technical and Economic Perspectives

Kenneth Q. Zhou, Johnny Siu-Hang Li

December 29, 2014

Abstract

In this paper, we study the feasibility of dynamic longevity hedging with standardized

securities that are linked to broad-based mortality indexes. On the technical front, we gen-

eralize the dynamic ‘delta’ hedging strategy developed by Cairns (2011) to incorporate the

situation when the populations associated with the hedger’s portfolio and the hedging instru-

ments are different. Our empirical results indicate that dynamic hedging can effectively reduce

the longevity exposure of a typical pension plan, even if population basis risk is taken into

account. On the economic front, we investigate the potential financial benefits of a dynamic

index-based hedge over a bespoke risk transfer. By considering data from a large group of na-

tional populations, we found evidence supporting the diversifiability of population basis risk.

It follows that for hedgers who intend to completely eliminate their longevity risk exposures,

it may be more economical to hedge the underlying trend risk with a dynamic index-based

hedge and transfer the residual basis risk through a reinsurance mechanism.

1 Introduction

The market for longevity risk transfers started in about 10 years ago. Since then, the market has

seen some significant developments, most notably in terms of the number and size of deals (Blake

et al., 2014). However, relative to the size of the global longevity risk exposure, the present

longevity risk transfer market is still very small. A small market not only impedes longevity

risk management, but also poses systemic concerns, because when longevity risk is shifted from

the corporate sector to a limited number of (re)insurers, with global interconnections, there may

be systemic consequences in the case of a failure of a key player (Basel Committee of Banking

1

Supervision, 2013).

The underdevelopment of the longevity risk transfer market may be attributed to the marked

imbalance between demand and supply. To date, most of the longevity risk transfers executed are

insurance-based, typically in the form of pension buy-ins, pension buy-outs or bespoke longevity

swaps. While the insurance industry has the scope and financial stability to assume longevity risk,

it does not generate sufficient supply for acceptance of the risk because of its capacity constraints.

Using the assets for pension plans, in excess of 31 trillion USD, as a proxy for demand and the

assets of 2.6 trillion USD held by the global insurance industry to cover non-life risks as a proxy

for supply, Graziani (2014) concluded that the demand for acceptance of longevity risk exceeds

supply by a multiple of 10. Michealson and Mulholland (2014) also reached a similar conclusion

by comparing the potential increase in pension liabilities due to unforeseen longevity improvement

with the aggregate capital of the global insurance industry.

The growth of the longevity risk transfer market therefore depends highly on the creation of

supply, most likely by inviting participation from capital markets, which are capable of assuming a

larger portion of the longevity risk exposures from pension plans around the world.1 The longevity

asset class offers capital market investors a risk premium, plus potential diversification benefits due

to its very low correlation with other asset classes. However, drawing interest from such investors

requires the longevity risk transfer market to package the risk as standardized products that are

structured like typical capital market derivatives and linked to broad-based mortality indexes.

The act of standardization is important in part because it fosters the development of liquidity,

and in part because it removes the information asymmetry arising from the fact that hedgers

(pension plans) have better knowledge about the mortality experience of their own portfolios.

Towards the goal of standardization, the market for longevity risk transfers has to overcome

two technical challenges which discourage hedgers from using standardized hedging instruments.

The first challenge is to find out how standardized instruments can be used to form a hedge that

can eliminate a meaningful portion of the hedger’s longevity risk exposure. Hedging strategies

have to be developed so that hedgers know the type and notional amounts of hedging instruments

they need to acquire. The second challenge is to understand and more importantly mitigate the

residual risks that are left behind by a standardized, index-based longevity hedge. Of the residual

1According to Roxburgh (2011), the total value of the world’s financial stock, comprising equity market capital-

ization and outstanding bonds and loans, is 212 trillion USD at the end of 2010.

2

risks the most significant constituent is population basis risk, which arises from the difference in

future mortality improvements between the population associated with the hedger’s own portfolio

and the population(s) to which the standardized instruments are linked. However, as explained

below, the research questions on longevity hedging strategies and population basis risk are still

open.

A significant portion of the existing literature on longevity hedging strategies focuses on static

hedging (Cairns, 2013; Cairns et al., 2006b, 2014; Coughlan et al., 2011; Dowd et al., 2011;

Li and Hardy, 2011; Li and Luo, 2012). Static hedging strategies are generally subject to the

shortcoming of the need for long-dated hedging instruments. For example, in an illustrative static

hedge for a 30-year pension liability, Li and Luo (2012) used five securities, of which the longest

time-to-maturity is 25 years. Such long-dated securities do not seem appealing to capital market

investors. A few researchers including Cairns (2011), Dahl (2004), Dahl and Møller (2006), Dahl

et al. (2008) and Luciano et al. (2012) proposed dynamic longevity hedging strategies. Except

the work of Cairns (2011), the existing dynamic longevity hedging strategies were developed from

continuous-time models, which provide mathematical tractability but are not straightforward

to implement in practice. Further, although some existing static hedging strategies include an

adjustment for population basis risk (Dowd et al., 2011; Li and Hardy, 2011; Li and Luo, 2012),

none of the aforementioned dynamic longevity hedging strategies takes population basis risk into

account.

For the problem of population basis risk, researchers have recently contributed significantly to

the development of multi-population stochastic mortality models (Ahmadi and Li, 2014; Cairns

et al., 2011; Dowd et al., 2011; Hatzopoulos and Haberman, 2013; Jarner and Kryger, 2011; Li

and Hardy, 2011; Li and Lee, 2005; Yang and Wang, 2013; Zhou et al., 2013, 2014). Such models

can be regarded as a pre-requisite for understanding population basis risk, because they allow

users to gauge the range of possible mortality differentials between two related populations, with

biological reasonableness taken into consideration. Researchers have also introduced metrics for

quantifying population basis risk, for example, reduction in expected shortfall (Ngai and Sherris,

2011), reduction in portfolio variance (Coughlan et al., 2011; Li and Hardy, 2011) and minimal

required buffer (Stevens et al., 2011). However, to our knowledge, little attention has been paid

to how population basis risk can be mitigated.

In this paper, we attempt to address the limitations of the current literature by investigating

3

how a dynamic, index-based longevity hedge can be performed when population basis risk is

present and how the residual risks left behind by the hedge can be mitigated. One part of the

framework is a dynamic hedging strategy with which a pension plan can transfer the ‘trend risk’

(i.e., the risk surrounding the trend in longevity improvement) to capital markets, even if the

securities available are linked to a broad-based mortality index. Another part of the framework

is a specially designed reinsurance treaty, called a ‘customized surplus swap’, which transfers the

residual risks to a reinsurer who collectively manages the residual risks from the index-based

longevity hedges of various pension plans.2

The dynamic hedging strategy we propose is obtained by generalizing the dynamic ‘delta’

hedging strategy of Cairns (2011) to incorporate the situation when the populations associated

with the hedger’s portfolio and the hedging instruments are not the same. The generalization is

derived on the basis of a multi-population stochastic mortality model, under which the mortality

dynamics of different populations are non-trivially correlated. When implementing the proposed

hedging strategy, the hedger needs to hold one only hedging instrument at a time and the hedging

instrument can be shorter-dated. The former property helps the market to concentrate liquidity,

while the latter property better meets the appetite of capital market investors. Adding further

to the contribution of Cairns (2011) is a study of the robustness of the dynamic hedging strategy

relative to different factors including model risk, small sample risk and the properties of the

hedging instruments used.

The customized surplus swap we design eliminates all residual risks that are left behind by

the dynamic longevity hedge. Therefore, the combination of a dynamic longevity hedge and cus-

tomized surplus swap should produce the same hedge effectiveness as a typical bespoke longevity

swap. Using real mortality data from 25 different populations, we demonstrate that the residual

risks can potentially be diversified away when a reinsurer write customized surplus swaps with

a range of hedgers. A reinsurer should thus have a much larger capacity to write customized

surplus swaps than contracts such as pension buy-outs which involve significant systematic risk.

Overall, our proposed risk management framework is likely to be more economical than tradi-

tional longevity risk transfers that are entirely insurance-based, because in theory it is less costly

2A similar concept was mentioned by Cairns et al. (2008). In their set-up, hedgers transfer all their longevity

risk exposures by writing bespoke longevity swaps with a special purposed vehicle (SPV), and the SPV in turn

issues a standardized longevity bond which transfers the trend risk to the bondholders. The residual risks are borne

by the SPV manager.

4

to transfer the systematic trend risk through liquidly traded standardized securities than tailor-

made (re)insurance contracts.

The rest of this paper is organized as follows. Section 2 presents the technical details of

the proposed dynamic hedging strategy. Section 3 illustrates the proposed dynamic hedging

strategy and evaluates its robustness relative to various factors. Section 4 defines the proposed

customized surplus swap and demonstrates the diversifiability of the residual risks. Finally, Section

5 concludes the paper with some suggestions for future research.

2 The Dynamic Longevity Hedging Strategy

2.1 The Assumed Model

The dynamic hedging strategy requires an assumed stochastic mortality model, from which quan-

tities such as hedge ratios can be derived. In the single-population set-up of Cairns (2011), the

original Cairns-Blake-Dowd model (a.k.a. Model M5) was assumed. In our generalization, we

assume the augmented common factor (ACF) model proposed by Li and Lee (2005). The ACF

model concurrently models the mortality dynamics of multiple, say P , populations as follows:

ln(m(i)x,t) = a(i)x +BxKt + b(i)x k

(i)t + ε

(i)x,t, i = 1, . . . , P,

where m(i)x,t represents population i’s central rate of death at age x and in year t, a

(i)x is a parameter

indicating population i’s average level of mortality at age x, Kt is a time-varying index that is

shared by all P populations, k(i)t is a time-varying index that is specific to population i, parameters

Bx and b(i)x respectively reflect the sensitivity of ln(m

(i)x,t) to Kt and k

(i)t , and ε

(i)x,t is the error term

that captures all remaining variations. Following Li and Lee (2005), we estimate the ACF model

by the method of singular value decomposition.

The trend in Kt determines the evolution of mortality over time for all populations being

modeled. As in the original Lee-Carter model (Lee and Carter, 1992), Kt is assumed to follow a

random walk with drift: Kt = C +Kt−1 + ξt, where C is the drift term and {ξt} is a sequence of

i.i.d. normal random variables with zero mean and constant variance σ2K .

Departures from the common time trend are captured by the population-specific index k(i)t ,

which is assumed to follow a first order autoregressive process: k(i)t = φ

(i)0 + φ

(i)1 k

(i)t−1 + ζ

(i)t ,

where φ(i)0 and φ

(i)1 are constants, and {ζ(i)t } is a sequence of i.i.d. normal random variables

5

with zero mean and constant variance σ2k,i. We require |φ(i)1 | < 1 so that the process for k(i)t

is mean-reverting. This property ensures that the resulting forecasts are coherent, which means

the projected mortality rates for different populations do not diverge indefinitely over time. To

incorporate any correlation that is not captured by the common trend Kt, we further assume

that ζ(i)t and ζ

(j)t for i 6= j are constantly correlated, despite such correlations are not taken into

account in the original ACF model.

2.2 The Set-up

We let

S(i)x,t(T ) =

T∏s=1

(1− q(i)x+s−1,t+s) (2.1)

be the ex post probability that an individual who is from population i and aged x at time t (the

end of year t) would have survived to time t + T , where q(i)x,t denotes the probability that an

individual from population i dies between time t− 1 and t (during year t), provided that he/she

has survived to age x at time t − 1. When computing q(i)x,t from m

(i)x,t (on which the ACF model

is based), we use the approximation q(i)x,t ≈ 1 − exp(−m(i)

x,t). It is clear from the definitions that

S(i)x,t(T ) is not known prior to time t+ T , while q

(i)x,t is not known prior to time t.

Define by Ft the information about the evolution of mortality up to and including time t. Due

to the Markov property of the assumed stochastic processes, the value of E(S(i)x,u(T )|Ft) for u ≥ t

depends only on the values of Kt and k(i)t but not the values of Kv and k

(i)v for v < t. Hence, we

have

p(i)x,u(T,Kt, k(i)t ) := E(S(i)

x,u(T )|Kt, k(i)t ) = E(S(i)

x,u(T )|Ft).

We call p(i)x,u(T,Kt, k

(i)t ) a spot survival probability when u = t and a forward survival probability

when u > t.

Let us suppose that the hedger intends to hedge the longevity risk associated with a pension

plan for a single cohort of individuals, who are all from population H and aged x0 at time 0. The

plan pays each pensioner $1 at the end of each year until death. It follows that the time-t value of

the pension plan’s future liabilities (per surviving pensioner at time t) can be expressed in terms

of spot survival probabilities as

FLt =

∞∑s=1

(1 + r)−s p(H)x0+t,t(s,Kt, k

(H)t ),

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where r is the interest rate for discounting purposes.

The hedging instruments are q-forwards that are associated with population R. A q-forward is

a zero-coupon swap with its floating leg proportional to the realized death probability at a certain

reference age during the year immediately prior to maturity and its fixed leg proportional to

the corresponding pre-determined forward mortality rate. In this application, the hedger should

participate in the q-forwards as the fixed-rate receiver, so that he/she will receive a net payment

from the counterparty when mortality turns out to be lower than expected.

Consider a q-forward that is linked to reference population R and age xf . Suppose that the

q-forward is issued at time t0 and matures at time t0+T ∗. The payoff from the q-forward depends

on the realized value of q(R)xf ,t0+T ∗ . The corresponding forward mortality rate qf is chosen so that

no payment exchanges hands at inception (time t0). It is assumed that qf = E(q(R)xf ,t0+T ∗ |Ft0),

which is equivalent to saying that no risk premium is given to the counterparty accepting the

risk.3 At t = t0, . . . , t0 + T ∗ − 1, the value of the hedger’s position of the q-forward (per $1

notional) can be expressed as

Qt(t0) = (1 + r)−(t0+T ∗−t)(qf − E(q(R)xf ,t0+T ∗ |Ft))

= (1 + r)−(t0+T ∗−t)(qf − (1− E(S(R)xf ,t0+T ∗−1(1)|Ft)))

= (1 + r)−(t0+T ∗−t)(qf − (1− p(R)xf ,t0+T ∗−1(1,Kt, k

(R)t )).

Under our pricing assumption, we have Qt0(t0) = 0. Note that both FLt and Qt(t0) are related

linearly to values of p(i)x,u(T,Kt, k

(i)t ), where i = H,R and u ≥ t.

The main idea behind the dynamic hedging strategy is that at each discrete time point t,

the q-forward portfolio is adjusted so that FLt and the adjusted q-forward portfolio have similar

sensitivities to changes in the underlying common mortality index Kt. Hence, at each discrete

time point t, we need to compute FLt and Qt(t0) and their partial derivatives with respect to Kt.

However, because of the way in which S(i)x,t(T ) depends on Ku and k

(i)u for u = t + 1, . . . , T , the

values of p(i)x,u(T,Kt, k

(i)t ) for u ≥ t (and thus FLt and Qt(t0)) cannot be computed analytically.

It follows that nested simulations are required, making the dynamic hedging framework strategy

computationally challenging.

3Because the counterparty accepting longevity risk from the hedger deserves a risk premium, in practice qf should

be smaller than E(q(R)xf ,t0+T∗ |Ft0), so that payoff to the counterparty is positive in expectation terms. However,

because our focus for now is the technical aspects rather than the associated costs, we assume qf = E(q(R)xf ,t0+T∗ |Ft0)

for simplicity.

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In more detail, let us assume that the hedging horizon is Y years and that the q-forward

portfolio is adjusted annually. Suppose that N sample paths of future mortality (i.e., values of

Kt, k(H)t and k

(R)t for t = 1, . . . , Y ) are used to evaluate the hedge’s performance. For each of

these N sample paths, we need to evaluate, at each time point t for t = 1, . . . , Y , FLt and Qt(t0)

on the basis of the realized values of Kt, k(H)t and k

(R)t in that particular sample path. If we

calculate each FLt and Qt(t0) with M sample paths of mortality beyond time t, then in total

we need to generate N ×M × Y sample paths. Because N and M are typically very large, say

10,000, the computational burden is huge. To reduce computation burden, in the next subsection

we derive formulas to approximate p(i)x,u(T,Kt, k

(i)t ) for u ≥ t so that the need for some of the

simulations can be avoided.

2.3 The Approximation Methods

The approximation formula for p(i)x,u(T,Kt, k

(i)t ) depends on whether u = t or u > t.

2.3.1 Approximating p(i)x,u(T,Kt, k

(i)t ) when u = t

Following Cairns (2011), we approximate p(i)x,t(T,Kt, k

(i)t ) by applying a Taylor expansion to its pro-

bit transform, f(i)x,t(T,Kt, k

(i)t ) := Φ−1(p

(i)x,t(T,Kt, k

(i)t )), where Φ denotes the standard normal dis-

tribution function. The Taylor expansion is made around K̂t = E(Kt|K0) and k̂(i)t = E(k

(i)t |k

(i)0 ).

We consider a second-order approximation, which means

f(i)x,t(T,Kt, k

(i)t ) ≈ D(i)

x,t,0(T ) +D(i)x,t,1(T )(Kt − K̂t) +D

(i)x,t,2(T )(k

(i)t − k̂

(i)t )

+1

2D

(i)x,t,3(T )(Kt − K̂t)

2 +1

2D

(i)x,t,4(T )(k

(i)t − k̂

(i)t )2

+D(i)x,t,5(T )(Kt − K̂t)(k

(i)t − k̂

(i)t ),

where

D(i)x,t,0(T ) = f

(i)x,t(T, K̂t, k̂

(i)t ), D

(i)x,t,1(T ) =

∂f(i)x,t(T,Kt,k̂

(i)t )

∂Kt

∣∣∣∣Kt=K̂t

,

D(i)x,t,2(T ) =

∂f(i)x,t(T,K̂t,k

(i)t )

∂k(i)t

∣∣∣∣k(i)t =k̂

(i)t

, D(i)x,t,3(T ) =

∂2f(i)x,t(T,Kt,k̂

(i)t )

∂K2t

∣∣∣∣Kt=K̂t

,

D(i)x,t,4(T ) =

∂2f(i)x,t(T,K̂t,k

(i)t )

∂k2,(i)t

∣∣∣∣k(i)t =k̂

(i)t

, D(i)x,t,5(T ) =

∂2f(i)x,t(T,Kt,k

(i)t )

∂Kt∂k(i)t

∣∣∣∣Kt=K̂t,k

(i)t =k̂

(i)t

.

The values of D(i)x,t,j(T ) for j = 1, . . . , 5 are computed numerically by finite difference ap-

proximations. For a fixed t, the finite difference approximations require nine sets of M sample

8

mortality paths, which are respectively based on nine different sets of starting values, including

(Kt = K̂t, k(i)t = k̂

(i)t ), (Kt = K̂t + h, k

(i)t = k̂

(i)t ), (Kt = K̂t, k

(i)t = k̂

(i)t + h) and so on. For a

hedging horizon of Y time steps, the number of sample paths required to generate the partial

derivatives is 9×M × Y .

Suppose again that N mortality scenarios are used to evaluate the hedge’s performance. Be-

cause the partial derivatives are independent of these N mortality scenarios, the total number of

sample paths we need to generate is N+9×M×Y , which is significantly smaller than N×M×Y

when N and M are large.

2.3.2 Approximating p(i)x,u(T,Kt, k

(i)t ) when u > t

Using a first-order approximation, it can be shown that

p(i)x,u(T,Kt, k(i)t ) ≈ Φ

−E(V(i)u |Ft)√

Var(V(i)u |Ft)

.

A proof of the above approximation formula and the expressions for E(V(i)u ) and Var(V

(i)u |Ft) are

provided in Appendix A. The accuracy of the two approximation formulas has been confirmed by

using contour plots that are similar to those presented in Cairns (2011).

2.4 Deriving Hedge Ratios

Our goal is to ensure that at each discrete time point t, the q-forward portfolio and the pension

plan’s future liabilities have similar sensitivities to changes in the underlying common mortality

index Kt. To achieve this goal, the hedge ratio ht (i.e., the notional amount of the q-forward) at

time t is chosen in such a way that

∂FLt

∂Kt= ht

∂Qt(t0)

∂Kt.

Because we match the first derivatives only, only one q-forward contract is needed at each t. For

the same reason, our hedge may be considered as a ‘delta’ hedge. In principle, one may create,

for example, a ‘gamma’ hedge by matching also the second order derivatives.

9

The partial derivative of FLt with respect to Kt is computed as follows:

∂FLt

∂Kt=∞∑s=1

(1 + r)−s∂p

(H)x0+t,t(s,Kt, k

(H)t )

∂Kt=∞∑s=1

(1 + r)−s∂Φ(f

(H)x0+t,t(s,Kt, k

(H)t ))

∂Kt

≈∞∑s=1

(1 + r)−sD(H)x0+t,t,1(s)φ(f

(H)x0+t,t(s,Kt, k

(H)t )),

where φ represents the probability density function for a standard normal random variable.

The partial derivative of Qt(t0) with respect to Kt depends on the value of t relative to the

q-forward’s maturity date t0 + T ∗. If t = t0 + T ∗ − 1,

∂Qt(t0)

∂Kt= (1 + r)−1

∂p(R)xf ,t

(1,Kt, k(R)t )

∂Kt≈ (1 + r)−1D

(R)xf ,t,1

(1)φ(f(R)xf ,t

(1,Kt, k(R)t )).

If t = t0, . . . , t0 + T ∗ − 2,

∂Qt(t0)

∂Kt=(1 + r)−(t0+T ∗−t)

∂p(R)xf ,t0+T ∗−1(1,Kt, k

(R)t )

∂Kt

≈(1 + r)−(t0+T ∗−t) ∂

∂KtΦ

−E(V(R)t0+T ∗−1|Ft)√

Var(V(R)t0+T ∗−1|Ft)

=(1 + r)−(t0+T ∗−t)φ

−E(V(R)t0+T ∗−1|Ft)√

Var(V(R)t0+T ∗−1|Ft)

−D(R)xf ,t0+T ∗−1,1(1)√

Var(V(R)t0+T ∗−1|Ft)

.

2.5 Evaluating the Hedge

As previously mentioned, N mortality scenarios are simulated to evaluate the effectiveness of the

dynamic hedge.

Define by PLt the time-0 value of all pension liabilities, given the information up to and

including time t; that is,

PLt = E

( ∞∑s=1

(1 + r)−sS(H)x0,0

(s)

∣∣∣∣Ft

)

=

FL0, t = 0∑ts=1(1 + r)−sS

(H)x0,0

(s) + (1 + r)−tS(H)x0,0

(t)FLt, t > 0.(2.2)

The value of PL0 is non-random, as it is simply a function of K0 and k(H)0 whose values are

fixed. For t > 0, the values of PLt are different under different simulated mortality scenarios. In

particular, the values of S(H)x0,0

(s) for s = 1, . . . , t depend on the realized values of Ks and k(H)s for

s = 1, . . . , t, whereas the value of FLt depends on the realized values of Kt and k(H)t .

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It is assumed that at each time point t, the hedger writes a new q-forward contract (i.e., a

q-forward with inception date t0 = t) with a notional amount of ht. The value of this position is

htQt(t) = 0 at time t and becomes

htQt+1(t) = ht(1 + r)−(T∗−1)(qf − E(q

(R)xf ,t+T ∗ |Ft+1))

= ht(1 + r)−(T∗−1)(E(q

(R)xf ,t+T ∗ |Ft)− E(q

(R)xf ,t+T ∗ |Ft+1))

= ht(1 + r)−(T∗−1)(p

(R)xf ,t+T ∗−1(1,Kt+1, k

(R)t+1)− p

(R)xf ,t+T ∗−1(1,Kt, k

(R)t ))

at time t+1.4 At time t+1, the position written at time t is closed out, and another new q-forward

contract is written. The process repeats until the end of the hedging horizon Y is reached. For

simplicity, we assume that all q-forwards used over the hedging horizon have the same maturity

T ∗ and reference age xf .

Let PAt be the time-0 value of the assets backing the pension plan at time t. We assume that

PA0 = PL0. For t = 1, . . . , Y , we have

PAt = PAt−1 + (1 + r)−tht−1Qt(t− 1).

If PAt is very close to PLt for t = 1, . . . , Y , then the dynamic hedge can be said as successful.

The potential deviation between PAt and PLt is the residual risk that is not mitigated by the

hedge. Using this reasoning, we measure hedge effectiveness by the following metric:

HEt = 1− Var(PAt − PLt|F0)

Var(PLt|F0).

A value of HEt that is close to one indicates the hedge is effective.

3 Analyzing the Dynamic Longevity Hedge

3.1 Assumptions

The following assumptions are used in the baseline calculations.

1. The hedger wishes to hedge the pension liabilities that are payable to a single cohort of

individuals, who are all aged x0 = 60 at time 0. The mortality experience of these individuals

is identical to that of the UK male insured lives.

4The second step is due to our pricing assumption.

11

2. The pension plan pays each individual $1 at the end of each year until death or age 90,

whichever the earliest.5

3. The hedging horizon is Y = 30 years (i.e., the hedge stops when the liabilities have com-

pletely run off).

4. The q-forwards used are linked to English and Welsh (EW) male population. They all have

a time-to-maturity (from inception) of T ∗ = 10 years and a reference age of xf = 75.

5. The market for the q-forwards is liquid and no transaction cost is required.

6. The interest rate for all durations is r = 4% per annum. The interest rate remains constant

over time.

7. The hedger can invest or borrow at an interest rate of r = 4% per annum.

8. The values of D(i)x,t,j(T ) for i = H,R and j = 0, . . . , 5 are computed from an ACF model

that is estimated to the data from the populations of EW males and UK male insured lives

over the period of 1966 to 2005 and the age range of 60 to 89.6

9. To match the end point of the data sample period, time 0 is set to the end of year 2005.

10. The evaluation of hedge effectiveness is based on N = 10, 000 mortality scenarios that are

generated from the model described in Assumption (8).

11. There is no small sample risk.

3.2 Baseline Results

The gray (larger) fan chart in the left panel of Figure 1 depicts the distributions of (PLt−PL0)|F0

(or equivalently PLt|F0) for t = 0, . . . , 30 under the baseline assumptions. When there is no

longevity hedge, the time-0 value of the assets backing the pension plan is always FL0, because

we assume PA0 = PL0 = FL0. Hence, PLt − PL0 can be regarded as the shortfall in assets in

the absence of a longevity hedge.

5We assume that no pension is payable beyond age 90, because the upper limit of the age range to which the

ACF model is fitted is 89. This assumption may be relaxed if one assumes a parametric curve to extrapolate death

probabilities beyond age 89.6The data for EW males are provided by the Human Mortality Database (2014), while the data for UK male

insured lives are obtained from the Institute and Faculty of Actuaries by a written request.

12

0 5 10 15 20 25 30−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time

Val

ue

With basis risk; HE30

= 0.9032

0 5 10 15 20 25 30−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time

Val

ue

Without basis risk; HE30

= 0.9997

Figure 1: Fan charts showing the distributions of (PLt−PL0)|F0 (in gray) and (PLt−PAt)|F0 (in green)

when population basis risk is present (the left panel) and hypothetically absent (the right panel).

The green (smaller) fan chart in the left panel Figure 1 shows the distributions of (PLt −

PAt)|F0 for t = 0, . . . , 30 under the baseline assumptions. We can regard PLt − PAt as the

shortfall in assets when a dynamic longevity hedge is in place. Over the entire hedging horizon,

(PLt−PAt)|F0 is significantly less dispersed than (PLt−PL0)|F0, indicating that the longevity

hedge is effective. Over the hedging horizon, the value of HEt is consistently larger than 90%.

To assess the extent of population basis risk, we repeat the calculations by assuming, hypo-

thetically, that q-forwards linked to the population of UK male insured lives are available and

used. The hedging results are shown in the right panel of Figure 1. The degree of population

basis risk can be observed from the difference in the widths of the green fan charts in left and

right panels of Figure 1.

3.3 Robustness

3.3.1 Robustness Relative to Model Risk

We first study how hedge effectiveness may change when the actual underlying model is not the

ACF model on which valuation and calculation of hedge ratios are based. To mimic this situation,

we use an alternative stochastic model to generate the N mortality scenarios for assessing hedge

effectiveness, while the ACF model is still used for valuation and calculation of hedge ratios. The

following two alternative models are considered.

13

• An asymmetric multi-population Lee-Carter model (M-LC)

Originally proposed by Cairns et al. (2011), the M-LC model has the following structure:

ln(m(i)x,t) = α(i)

x + βxκ(i)t + e

(i)x,t, i = 1, . . . , P,

where α(i)x and βx are age-specific parameters, κ

(i)t is a time-varying parameter and e

(i)x,t

is the error term. The model is considered as asymmetric, because one population being

modeled (say population id) is assumed to be dominant, driving the mortality dynamics of

the other populations. The evolution of κ(id)t over time is modeled by a random walk with

drift, while the differential κ(id)t − κ(i)t for i 6= id is modeled by a first order autoregressive

process. These processes ensure that the resulting forecast is coherent. In our illustration,

we assume that the dominant population is EW males.

• A multi-population Cairns-Blake-Dowd model (M-CBD)

The M-CBD model is an extension of the original Cairns-Blake-Dowd model (Cairns et al.,

2006a). It can be expressed as

ln

(q(i)x,t

1− q(i)x,t

)= κ∗1,t + κ∗2,t(x− x̄) + κ

(i)1,t + κ

(i)2,t(x− x̄) + e

(i)x,t, i = 1, . . . , P,

where x̄ denotes the average age over the sample age range, κ∗1,t and κ∗2,t are time-varying pa-

rameters that are shared by all P populations, κ(i)1,t and κ

(i)2,t are time-varying parameters that

apply only to population i, and e(i)x,t is the error term. The vector of κ∗1,t and κ∗2,t is modeled

by a bivariate random walk with drift. Each κ(i)1,t is modeled by a first order autoregression,

with a mean-reverting property that ensures the resulting projection is coherent.

The values of HE30 under different simulation models are shown in the section labeled ‘Sensi-

tivity Test 1’ in Table 1. The hedging result when the simulation model is M-LC is quite close to

the baseline result. This outcome may be attributed to the fact that the ACF and M-LC models

are similar. Both models are generalizations of the single-population Lee-Carter model, and both

models contain only one time-varying factor that is shared by all populations being modeled.

Also, as the M-LC model contains one less stochastic process than the ACF model,7 it may imply

less stochastic uncertainty, which may explain why it leads to a hedging result that is even better

than the baseline result.

7In this application, the ACF model contains three stochastic processes (one for Kt, one for k(H)t and one for

k(R)t ), whereas the M-LC model contains two stochastic processes (one for κ

(R)t and another for κ

(H)t − κ(R)

t ).

14

Sensitivity Test 1Simulation Model ACF M-LC M-CBD

HE30 90.28% 95.09% 90.27%

Sensitivity Test 2l60 ∞ 10,000 3,000 1,000

HE30 90.28% 86.19% 77.23% 60.12%

Sensitivity Test 3xf 65 70 75 85

HE30 94.39% 93.91% 90.28% 90.51%

Sensitivity Test 4T ∗ 5 10 15 20

HE30 81.79% 90.28% 92.22% 92.78%

Table 1: The values of HE30 calculated under different assumptions

The hedging result when the simulation model is M-CBD is close to but not as good as the

baseline result. This outcome may be explained by the fact that the M-CBD model contains two

stochastic factors that are common to both populations, but the hedge is composed of only one

instrument at a time. Nevertheless, the value of HE30 produced under this simulation model is

still above 90%, indicating that the hedge remains highly effective even if the true underlying

model is different and more sophisticated.

3.3.2 Robustness Relative to Small Sample Risk

Next, we investigate the impact of small sample risk (a.k.a. sampling risk and Poisson risk) on

hedge effectiveness. The cohort of pensioners is now treated as a random survivorship group, so

that given the values of lx0+s−1 and q(H)x0+s−1,s,

lx0+s ∼ Binomial(lx0+s−1, 1− q(H)x0+s−1,s),

s = 1, . . . , Y , where lx represents the number of pensioners who survive to age x. Note that lx0 is

non-random.

The procedure and assumptions for calculating hedge effectiveness remain the same, except

that the values of S(H)x0,0

(s), s = 1, . . . , t, in PLt are now calculated with an additional simulation

routine: for each of the N mortality scenarios generated, we simulate a realization of lx0+s using

the above binomial process, and then calculate the realized value of S(H)x0,0

(s) as l̃x0+s/lx0 , where

l̃x0+s denotes the realized value of lx0+s. Because small sample risk affects only the pension plan’s

realized mortality experience, the values of FLt, Qt(t) and ht are unaffected.

15

In the section labeled ‘Sensitivity Test 2’ in Table 1 we show the hedging results when the

pension plan begins at time 0 with l60 = ∞, 10,000, 3,000 and 1,000 individuals aged x0 = 60.

The hedge effectiveness is still very high (HE30 is close to 90%) when l60 = 10, 000, but the

impact of small sample risk becomes apparent as l60 reduces to 3,000. These observations are in

line with the results produced by Li and Hardy (2011) who considered a static longevity hedge.

3.3.3 Robustness Relative to the q-Forwards’ Reference Age

In early stages of market development, the availability of q-forwards is likely to be limited. It is

therefore important to understand how hedge effectiveness may change if the characteristics of

the q-forwards used are different.

We hereby test the robustness of the hedge effectiveness relative to the reference age xf of

the q-forwards used. The section labeled ‘Sensitivity Test 3’ in Table 1 shows the hedging results

when xf = 65, 70, 75, 80. It can be seen that changes in xf have only a negligible effect on the

hedging result. For all four choices of xf , the values of HE30 are over 90%.

Recall that the dynamic longevity hedge is constructed by matching the sensitivities of the

pension plan’s liabilities and the hedge portfolio with respect to Kt. Therefore, a hedging instru-

ment tends to be effective if its payoff is heavily dependent on the randomness associated with Kt

(which affects both the hedging instrument and the liabilities being hedged) but not so much on

the randomness associated with k(H)t (which affects the hedging instrument but has little effect

on the liabilities being hedged). In particular, for a q-forward with reference age xf , the resulting

hedge effectiveness tends to be high if

Var(BxfKt+T ∗ |Ft)� Var(b(H)

xfk(H)t+T ∗ |Ft).

Given the parameter estimates, we have Bxf� b

(H)xf for xf = 65, . . . , 80 and Var(Kt+T ∗ |Ft) =

T ∗σ2K � Var(k(H)t+T ∗ |Ft) = (1− (φ

(H)1 )2T

∗)σ2k,H/(1− (φ

(H)1 )2) for T ∗ = 10. Therefore, the relation

above holds and the hedging results are generally good. The four choices of xf lead to slightly

different hedging results, because there exist small variations in the estimates of Bx and b(H)x over

the age range of 65 to 80.

16

3.3.4 Robustness Relative to the q-Forwards’ Time-to-Maturity

Finally, we study the robustness of the hedge effectiveness relative to the time-to-maturity T ∗ of

the q-forwards used. We implement the dynamic longevity hedge using q-forwards with maturities

of 5, 10, 15 and 20 years. The hedging results are displayed in the section labeled ‘Sensitivity

Test 4’ in Table 1.

The dynamic longevity hedge is more effective when the q-forwards used have a longer time-to-

maturity. This result is because as T ∗ increases, Var(Kt+T ∗ |Ft) grows linearly while Var(k(H)t+T ∗ |Ft)

approaches gradually to a constant, which in turn means that the random underlying mortality

rate becomes relatively more dependent on the randomness associated with Kt (which affects both

the q-forward and the liabilities being hedged) but less on the randomness associated with k(H)t

(which has little effect on the liabilities being hedged).

Still, even when T ∗ is as small as five years, the value of HE30 is higher than 80%. The high

effectiveness can be attributed to the dynamic nature of our hedging strategy. Because we adjust

the hedge annually and hold each q-forward for only one year, each q-forward is responsible for

hedging the uncertainty that is one year ahead only. For this reason, short-dated q-forwards still

lead to highly satisfactory results, despite the liability payments last for 30 years. This feature

distinguishes our method from static hedging strategies, such as that proposed by Li and Luo

(2012), which generally require longer-dated instruments to achieve a satisfactory result.

4 Managing the Residual Risks

In this section, we explain how the residual risks from a dynamic, index-based longevity hedge

can be managed through a reinsurance mechanism.

4.1 Assumptions

As we expand our analysis to include more than two populations, some of the previously made

assumptions have to be modified accordingly. Below we list the assumptions that are used in this

section.

(i) There are 25 pension plans wishing to hedge their longevity risk exposures. The 25 pension

plans have respectively identical mortality experience to 25 different male populations. The

17

chosen populations are the same as the 25 populations that are classified as the ‘males

West-cluster’ by Hatzopoulos and Haberman (2013).

(ii) Each pension plan contains initially l60 = 3, 000 pensioners who are all aged x0 = 60. For

x = 61, 62, . . ., lx follows the binomial process described in Section 3.3.2.

(iii) At any time point during the hedging horizon, the only hedging instrument available is a

q-forward that is linked to EW male population with a time-to-maturity (from inception)

of T ∗ = 10 years and a reference age of xf = 75.

(iv) The values of D(i)x,t,j(T ) for i = 1, . . . , 25 and j = 0, . . . , 5 are computed from an ACF model

that is estimated to the data from the 25 male populations under consideration.8

(v) To match the end point of the data sample period, time 0 is set to the end of year 2009.

(vi) The evaluation of hedge effectiveness is based on N = 10, 000 mortality scenarios that are

generated from the model described in Assumption (iv).

Assumptions (2), (3), (5), (6) and (7) stated in Section 3.1 remain unchanged.

4.2 A Customized Surplus Swap

A dynamic longevity hedge is implemented for each of the 25 pension plans. The hedging results

vary significantly, with HE30 ranging from 38% to 80%. The results indicate that the dynamic

longevity hedge may leave substantial residual risks, which include small sample risk and popu-

lation basis risk.

In what follows, we propose a customized surplus swap that permits pooling of the residual

risks from different dynamic longevity hedges. When implementing such a swap in tandem with a

dynamic index-based hedge, the pension plan would in theory be immunized from longevity risk.

A pension plan is immunized from longevity risk over the hedging horizon if PAt−PLt = 0 for

t = 1, . . . , Y . We can regard |PAt−PLt| as the pension plan’s surplus if PAt > PLt and short fall

in assets if PAt < PLt. The swap we design has a maturity of one year and is written at each time

point when the dynamic index-based hedge is established or adjusted. We call it a ‘surplus’ swap,

8The mortality data for all 25 male populations are obtained from the Human Mortality Database (2014). The

data used cover a sample period of 1959 to 2009 and a sample age range of 60 to 89.

18

because its net cash flow at maturity is derived from the surplus process PAt−PLt, t = 1, . . . , Y ,

of the pension plan.

Our goal is to ensure that PAt−PLt = 0 for t = 1, . . . , Y . We let NCFt be the net cash flow

(payable at time t from the reinsurer to the pension plan) for the customized surplus swap that is

written at time t− 1. With the swap in place, the recursion formula for PAt can be rewritten as

PAt = PAt−1 + (1 + r)−t(ht−1Qt(t− 1) +NCFt), t = 1, . . . , Y, (4.1)

where PA0 = PL0. Using equations (4.1) and (2.2), we obtain

PLt − PAt = PLt−1 − PAt−1 + (1 + r)−t(S(H)x0,0

(t)(1 + FLt)− (1 + r)S(H)x0,0

(t− 1)FLt−1)

− (1 + r)−t(ht−1Qt(t− 1) +NCFt), t = 1, . . . , Y.

To stipulate PAt − PLt = 0 for t = 1, . . . , Y , we require

NCFt = S(H)x0,0

(t)(1 + FLt)− ht−1Qt(t− 1)− (1 + r)S(H)x0,0

(t− 1)FLt−1, t = 1, . . . , Y.

The expression for NCFt is intuitive. It says that there is no net cash flow from the swap if

what the pension plan has at time t− 1 accumulated with interest (i.e., (1 + r)S(H)x0,0

(t− 1)FLt−1)

plus the proceed from the index-based hedge at time t (i.e., ht−1Qt(t − 1)) is just sufficient to

cover the plan’s financial obligations at time t (i.e., S(H)x0,0

(t)) and beyond (i.e., S(H)x0,0

(t)FLt).

Given Ft−1, the value of (1 + r)S(H)x0,0

(t−1)FLt−1 is known, but the values of S(H)x0,0

(t)(1 +FLt)

and ht−1Qt(t−1) are random as they both depend on the values of Kt, k(H)t and k

(R)t which are not

known until time t. It follows that for a customized surplus swap written at time t− 1, the fixed

and floating legs should be set to (1 + r)S(H)x0,0

(t− 1)FLt−1 and S(H)x0,0

(t)(1 +FLt)− ht−1Qt(t− 1),

respectively.

Given how the cash flows are defined, the following should be incorporated into the terms of

a customized surplus swap written at time t− 1:

• the method and assumptions used to calculate FLt−1 and FLt;

• the hedge ratio ht−1;

• the rate r at which the cash flows are discounted;

• the forward mortality rate qf associated with the q-forward written at time t− 1.

19

For simplicity, we assume that the swap is costless in the following illustration. In practice, of

course, the reinsurer demands a reward for taking on the risk and therefore a fixed payment (the

risk premium) has to be paid by the pension plan to the reinsurer at either inception or maturity.

4.3 An Illustration

We now revisit the index-based longevity hedges for the 25 pension plans. Let us suppose that on

top of the index-based hedges, all 25 pension plans write customized surplus swaps with the same

reinsurer to eliminate their exposures to the residual risks. Assume further that the assumptions

used in formulating the index-based hedges also apply to the terms of the customized surplus

swaps.

The fan charts in Figure 2 show the distributions of NCFt|F0 for the 25 pension plans. To

study the diversifiability of the residual risks from the index-based hedges, let us consider the

cash flows from the viewpoint of the reinsurer who writes customized surplus swaps with the 25

pension plans. The fan chart in the left panel of Figure 3 depicts the distributions (conditioned

on F0) of the average net cash flows payable to each pension plan over the hedging horizon. The

variability of the reinsurer’s average net cash flows is small compared to the variability of NCFt

for individual pension plans. The diversifiability can be observed more clearly from the right

panel of Figure 3, which compares the variances of the reinsurer’s average net cash flows with the

variances of the net cash flows arising from individual customized surplus swaps.

5 Discussion and Conclusion

In this paper, we consider a risk management framework in which longevity risk is split into trend

risk and residual risks. With the proposed dynamic hedging strategy, a pension plan can transfer

its trend risk exposure to capital markets through standardized instruments. Using the proposed

customized surplus swap, the pension plan may also transfer the residual risks left behind by the

dynamic hedge to a reinsurer, who collectively manages the residual risks from various pension

plans. As a whole, our risk management framework allows pension plans to completely eliminate

their longevity risk exposures, just as what they can achieve from traditional, entirely insurance-

based pension de-risking solutions.

What we propose allows the longevity risk transfer market to package the trend risk as stan-

20

0 15 30−0.2

0

0.2

NC

Ft

EW

0 15 30−0.2

0

0.2Scotland

0 15 30−0.2

0

0.2E. Germany

0 15 30−0.2

0

0.2W. Germany

0 15 30−0.2

0

0.2France

0 15 30−0.2

0

0.2

NC

Ft

Portugal

0 15 30−0.2

0

0.2Switzerland

0 15 30−0.2

0

0.2Belgium

0 15 30−0.2

0

0.2Finland

0 15 30−0.2

0

0.2Canada

0 15 30−0.2

0

0.2

NC

Ft

Austria

0 15 30−0.2

0

0.2Italy

0 15 30−0.2

0

0.2New Zealand

0 15 30−0.2

0

0.2Spain

0 15 30−0.2

0

0.2USA

0 15 30−0.2

0

0.2

NC

Ft

Luxembourg

0 15 30−0.2

0

0.2The Netherlands

0 15 30−0.2

0

0.2Sweden

0 15 30−0.2

0

0.2Ireland

0 15 30−0.2

0

0.2Norway

0 15 30−0.2

0

0.2

Time (t)

NC

Ft

Australia

0 15 30−0.2

0

0.2

Time (t)

Iceland

0 15 30−0.2

0

0.2

Time (t)

Japan

0 15 30−0.2

0

0.2

Time (t)

Czech

0 15 30−0.2

0

0.2

Time (t)

Denmark

Figure 2: Fan charts showing the distributions of NCFt|F0 for the 25 pension plans under consideration.

dardized products that are structured like typical capital market derivatives. Compared to prod-

ucts such as pension buy-ins, standardized mortality derivatives are more appealing to capital

market investors who generally desire liquidity and transparency. When put in practice, our risk

management framework may attract participation from capital markets, thereby ameliorating the

demand and supply imbalance in the present market for longevity risk transfers.

The enhancement of liquidity through standardization may also result in lower risk man-

agement costs to pension plans, as the illiquidity premium payable to the counterparty can be

reduced. Although there is no sufficient data to test the inverse relationship between liquidity and

compensation to investors (typically measured by the Sharpe ratio) in the longevity risk market,

there is profound evidence for such an inverse relationship in markets for common stocks (Lo

et al., 2003), mutual funds (Idzorek et al., 2012) and hedge funds (Getmansky et al., 2004). It

has been argued that the market for longevity risk transfers has many similarities compared to a

typical financial market (Loeys et al., 2007). Hence, it is reasonable to conjecture that the inverse

relationship between liquidity and Sharpe ratio found in other markets also applies to the market

for longevity risk transfers. Should this conjecture holds, then our risk management framework

would be more economical than the comparable entirely insurance-based methods, because it

21

0 5 10 15 20 25 30−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time

Ave

rage

net

cas

h flo

w

5 10 15 20 25 300

1

2

3

4

5

6

7x 10

−3

Time

Var

ianc

e

Figure 3: An illustration of the diversifiability of the residual risks from the index-based hedges. The

left panel shows the distributions (conditioned on F0) of the average net cash flows payable from the

reinsurer to each pension plan over the hedging horizon. The right panel shows the values of Var(NCFt|F0),

t = 1, . . . , 30, for the individual customized surplus swaps (the dotted lines) and the corresponding variances

of the reinsurer’s average net cash flows payable to each pension plan (the solid line).

could transfer the trend risk at a lower cost. A reduced cost may encourage more pension plans

to transfer their longevity risk exposures, thereby not only facilitating market growth but also

strengthening the stability of the pension industry.

To focus on the design and execution of the proposed risk management methods, we have made

no attempt to estimate the associated costs. It thus warrants a separate study to investigate how

much the proposed risk management methods may cost. To determine the cost associated with

the dynamic longevity hedge, one may replace qf with a forward mortality rate that is derived

from the pricing methods proposed by Chuang and Brockett (2014), Deng et al. (2012) and Li

et al. (2011). As a reinsurance treaty, the customized surplus swap may be priced under the

Solvency II framework. In particular, its profit margin may be calculated by multiplying the

present value of the solvency capital requirements with the spread over risk-free rate which the

reinsurer is required to earn on its equity (see Zhou et al., 2014). Also, to understand the value

for money of our risk management framework, it would be interesting to compare the total cost

required by the proposed risk management methods with that required by a full pension buy-out.9

As the proposed dynamic hedging strategy matches only the first partial derivatives (with

9Mercer provides pension buy-out indexes, which track the estimated cost of a full pension buy-out in the US,

the UK, Ireland and Canada over time.

22

respect to the common time trend Kt), it requires the hedger to hold only one hedging instrument

at a time. This property may be seen as advantageous, because it helps the market to concentrate

liquidity. In future research, it may be fruitful to extend the proposed dynamic hedging strategy

to match also the higher order derivatives, and to investigate whether such an extension would

lead to an improvement in hedge effectiveness that worths the dilution of liquidity arising from

the use of additional instruments.

The results of various robustness tests indicate that the effectiveness of the dynamic longevity

hedge is reasonably robust relative to model risk and the parameters of the q-forwards used. They

also offer some useful insights to market participants. For example, because the dynamic hedge

still yields satisfactory hedging results even if the time-to-maturity of the q-forwards is only five

years, the market may choose to launch shorter-dated q-forwards, which are more likely to attract

capital market investors. As robustness is important in gaining trust from various stakeholders,

we believe that future research warrants a more extensive analysis of robustness which considers

additional aspects of the longevity hedge (e.g., hedge ratios) and additional factors that may affect

hedge effectiveness (e.g., parameter risk).

In illustrating the customized surplus swap, mortality data from a group of distinct national

populations are used. In reality, however, a reinsurer may possibly write customized surplus swaps

with pension plans that are located in the same country, so it is also important to understand

the diversifiability of residual risks across sub-populations with the similar geographical locations

but different social-economic statuses. Such an understanding may be developed by considering

the Club Vita data set of UK occupational pension schemes that was used by Haberman et al.

(2014). We suggest revisiting the study of the customized surplus swap when the Club Vita data

set becomes available from the public domain.

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Appendix A

Deriving the Approximation Formula for p(i)x,u(T,Kt, k

(i)t ) when u > t

Let Z be a standard normal random variable that is independent of Kt and k(i)t . Using a first-order

approximation, we have

p(i)x,t(T,Kt, kt) ≈ Φ(D

(i)x,t,0(T ) +D

(i)x,t,1(T )(Kt − K̂t) +D

(i)x,t,2(T )(k

(i)t − k̂

(i)t ))

= Pr(Z ≤ D(i)x,t,0(T ) +D

(i)x,t,1(T )(Kt − K̂t) +D

(i)x,t,2(T )(k

(i)t − k̂

(i)t )|Kt, k

(i)t )

= E(IZ≤D(i)

x,t,0(T )+D(i)x,t,1(T )(Kt−K̂t)+D

(i)x,t,2(T )(k

(i)t −k̂

(i)t )|Kt, k

(i)t )

= E(IZ≤D(i)

x,t,0(T )+D(i)x,t,1(T )(Kt−K̂t)+D

(i)x,t,2(T )(k

(i)t −k̂

(i)t )|Ft),

26

where IA is an indicator function which equals 1 if event A holds and 0 otherwise. The last step

is due to the Markov property of the assumed stochastic processes for Kt and k(i)t .

For u > t, we have

p(i)x,u(T,Kt, k(i)t ) = E(p(i)x,u(T,Ku, k

(i)u )|Ft)

≈ E(E(IZ≤D(i)

x,u,0(T )+D(i)x,u,1(T )(Kt−K̂u)+D

(i)x,t,2(T )(k

(i)u −k̂

(i)u )|Fu)|Ft)

= E(IZ≤D(i)

x,u,0(T )+D(i)x,u,1(T )(Ku−K̂u)+D

(i)x,u,2(T )(k

(i)u −k̂

(i)u )|Ft)

= Pr(Z ≤ D(i)x,u,0(T ) +D

(i)x,u,1(T )(Ku − K̂u) +D

(i)x,u,2(T )(k(i)u − k̂(i)u )|Ft)

= Pr(Z −D(i)x,u,0(T )−D(i)

x,u,1(T )(Ku − K̂u)−D(i)x,u,2(T )(k(i)u − k̂(i)u ) ≤ 0|Ft).

Let V(i)u = Z−D(i)

x,u,0(T )−D(i)x,u,1(T )(Ku−K̂u)−D(i)

x,u,2(T )(k(i)u −k̂(i)u ). On the basis of the assumed

stochastic processes, Ku|Ft, k(i)u |Ft and thus V

(i)u |Ft are normally distributed. It immediately

follows that

p(i)x,u(T,Kt, k(i)t ) ≈ Φ

−E(V(i)u |Ft)√

Var(V(i)u |Ft)

,

where

E(V (i)u |Ft) = −D(i)

x,u,0(T )−D(i)x,u,1(T )(E(Ku|Ft)− K̂u)−D(i)

x,u,2(T )(E(k(i)u |Ft)− k̂(i)u ),

Var(V (i)u |Ft) = 1 + (D

(i)x,u,1(T ))2Var(Ku|Ft) + (D

(i)x,u,2(T ))2Var(k(i)u |Ft)

+ 2D(i)x,u,1(T )D

(i)x,u,2(T )Cov(Ku, k

(i)u |Ft).

Under the assumed stochastic processes, we have

E(Ku|Ft)− K̂u = Kt −K0 − Ct,

E(k(i)u |Ft)− k̂u = (φ(i)1 )u((φ

(i)1 )−tk

(i)t − k

(i)0 ) +

(φ(i)1 )u(1− (φ

(i)1 )−t)

1− φ(i)1

φ(i)0 ,

Var(Ku|Ft) = Var(Ku|Kt) = σ2K(u− t),

Var(k(i)u |Ft) = Var(k(i)u |k(i)t ) =

1− (φ(i)1 )2(u−t)

1− (φ(i)1 )2

σ2k,i,

and Cov(Ku, k(i)u |Ft) = 0.

27